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Let ($t_n$) be a sequence which converges to $0$: $$\lim_{n\to\infty} t_n=0 $$ Let ($s_n$) be a sequence which is bounded both from above and below. Show $$\lim_{n\to\infty} t_n s_n=0 $$

I'm not totally sure how to approach this problem. Because $s_n$ doesn't necessarily have a limit I've tried approaching it by bringing up the definitions of boundedness but I'm a little lost. What would be the best starting point to go for here?

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  • $\begingroup$ The boundedness condition just means $\forall n,\ a\le s_n\le b$ for some constants $a,b$. Also if $c$ is a constant $c\,t_n\to 0$. $\endgroup$ – zwim Oct 12 '17 at 4:46
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Squeeze Theorem. $$ \ \ \ \ \ \ \ \ \ \ $$

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  • $\begingroup$ So you'd suggest using this to prove that (sn) has a limit? I think I know how to solve it from there thanks! $\endgroup$ – Jebus Christo Oct 12 '17 at 4:12
  • $\begingroup$ No, $\{s_n\} $ does not have to be convergent. $\endgroup$ – Martin Argerami Oct 12 '17 at 5:21

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